A lad was asked his age by his friend. The lad said, "The number you get when you subtract 25 tiems my age from twice the square of my age will be thrice your age". If the friend's age is 14, then the age of the lad is:

Options

A. 21/2
B. 28
C. 14
D. 25

Answer

To solve the problem step by step, we will define the lad's age and set up the equation based on the information provided.

### Step 1: Define the variables
Let the age of the lad be \( x \).

### Step 2: Set up the equation
According to the problem, the lad states:
"The number you get when you subtract 25 times my age from twice the square of my age will be thrice your age."

We can express this mathematically:
- Twice the square of his age: \( 2x^2 \)
- 25 times his age: \( 25x \)
- Thrice the friend's age (given that the friend's age is 14): \( 3 \times 14 = 42 \)

So, the equation becomes:
\[ 2x^2 - 25x = 42 \]

### Step 3: Rearrange the equation
To form a standard quadratic equation, we rearrange it:
\[ 2x^2 - 25x - 42 = 0 \]

### Step 4: Solve the quadratic equation
We can solve this quadratic equation using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 2 \), \( b = -25 \), and \( c = -42 \).

Calculating the discriminant:
\[ b^2 - 4ac = (-25)^2 - 4 \cdot 2 \cdot (-42) \]
\[ = 625 + 336 \]
\[ = 961 \]

Now, applying the quadratic formula:
\[ x = \frac{-(-25) \pm \sqrt{961}}{2 \cdot 2} \]
\[ = \frac{25 \pm 31}{4} \]

Calculating the two possible values for \( x \):
1. \( x = \frac{25 + 31}{4} = \frac{56}{4} = 14 \)
2. \( x = \frac{25 - 31}{4} = \frac{-6}{4} = -1.5 \) (not a valid age)

Thus, the only valid solution is:
\[ x = 14 \]

### Step 5: Conclusion
The age of the lad is \( 14 \).